Metric Entropy of Homogeneous Spaces

• Autorzy: Stanisław J. Szarek
• Języki publikacji: EN

Abstrakty

EN

For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the metric entropy, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous spaces of unitary (or orthogonal) groups with respect to some natural metrics, most notably the one induced by the operator norm. This generalizes the author's earlier results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. The argument uses a characterization of geodesics in U(n) (or SO(m)) for a class of non-Riemannian Finsler metric structures.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1998
• Tom: 43
• Numer: 1
• Strony: 395-410

Daty

wydano

1998

Twórcy

autor

Stanisław J. Szarek

• Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106-7058, U.S.A.

Bibliografia

• [1] K. Ball, E. A. Carlen and E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math. 115 (1994), no. 3, 463-482.
• [2] R. Bhatia, Perturbation bounds for matrix eigenvalues, Pitman Research Notes 162, Longman Scientific & Technical, Harlow 1987.
• [3] H. Busemann, Metric methods in Finsler spaces and the foundations of geometry, Annals Math. Studies 8, Princeton University Press, Princeton, 1942.
• [4] B. Carl and I. Stephani, Entropy, Compactness and Approximation of Operators, Cambridge University Press, Cambridge, 1990.
• [5] S. Chevet, Séries de variables aléatoires gaussiennes à valeurs dans $E ^⊗_ε F$, Séminaire sur la géométrie des espaces de Banach 1977-78, Ecole Polytechnique, Palaiseau.
• [6] K. Dykema, personal communication.
• [7] L. Ge, Prime factors, Proc. Natl. Acad. Sci. USA 93 (1996), 12762-12763.
• [8] L. Ge, Applications of free entropy to finite von Neumann algebras, Amer. J. Math. 119 (1997), 467-485
• [9] E. D. Gluskin, The diameter of the Minkowski compactum is roughly equal to n, Functional Anal. Appl. 15 (1981), 72-73.
• [10] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Nauka, Moscow 1965. English transl.: AMS 1969.
• [11] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York 1978.
• [12] D. Herrero and S. J. Szarek, How well can an n × n matrix be approximated by reducible ones?, Duke Math. J. 53 (1986), 233-248.
• [13] A. Pajor, Metric entropy of the Grassmann manifold, Proceedings of the MSRI Convex Geometry Semester (Spring 1996), to appear.
• [14] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge University Press, Cambridge 1989.
• [15] J. Saint-Raymond, Sur le volume des idéaux d'opérateurs classiques, Studia Math. 80 (1984), 63-75.
• [16] L. A. Santaló, Un invariante afin para los cuerpos convexos del espacio de n dimensiones, Port. Math. 8(1949), 155-161.
• [17] S. J. Szarek, Nets of Grassmann manifolds and orthogonal groups, Proceedings of Banach Space Workshop, University of Iowa Press 1982, 169-185.
• [18] S. J. Szarek, The finite dimensional basis problem with an appendix on nets of Grassmann manifolds, Acta Math. 151 (1983), 153-179.
• [19] S. J. Szarek, An exotic quasidiagonal operator, J. Funct. Anal. 89 (1990), 274-290.
• [20] S. J. Szarek, Spaces with large distance to $l_{∞} ^{n}$ and random matrices, Amer. J. Math. 112 (1990), 899-942.
• [21] S. J. Szarek, Metric entropy of homogeneous spaces and Finsler geometry of classical Lie groups, MSRI preprint 1997-010.
• [22] S. J. Szarek, Geodesics for Invariant Finsler Geometries on Classical Lie Groups, to appear.
• [23] N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, Longman Scientific & Technical, Harlow 1989.
• [24] V. Varadarajan, Lie Groups, Lie Algebras and Their Representations, Springer Verlag, New York 1984.
• [25] D. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability theory. III. The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996), 172-199.
• [26] D. Voiculescu, personal communication.